Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 3 times and lost 7 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Goat
6. Car
7. Goat
8. Goat
9. Car
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 313 times and lost 687 times.

1. Car
2. Car
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
8. Car
9. Car
10. Car
11. Goat
12. Goat
13. Goat
14. Goat
15. Car
16. Goat
17. Goat
18. Car
19. Goat
20. Car
21. Goat
22. Goat
23. Goat
24. Goat
25. Goat
26. Goat
27. Goat
28. Goat
29. Goat
30. Goat
31. Car
32. Goat
33. Car
34. Goat
35. Car
36. Goat
37. Goat
38. Goat
39. Goat
40. Goat
41. Goat
42. Goat
43. Goat
44. Goat
45. Goat
46. Goat
47. Goat
48. Goat
49. Goat
50. Goat
51. Goat
52. Goat
53. Goat
54. Goat
55. Car
56. Goat
57. Car
58. Goat
59. Goat
60. Goat
61. Car
62. Car
63. Goat
64. Goat
65. Goat
66. Car
67. Car
68. Car
69. Goat
70. Car
71. Goat
72. Goat
73. Goat
74. Goat
75. Car
76. Car
77. Car
78. Car
79. Goat
80. Car
81. Goat
82. Goat
83. Goat
84. Goat
85. Car
86. Goat
87. Goat
88. Goat
89. Car
90. Goat
91. Goat
92. Goat
93. Goat
94. Goat
95. Car
96. Goat
97. Goat
98. Car
99. Goat
100. Car
101. Car
102. Car
103. Goat
104. Car
105. Car
106. Car
107. Car
108. Goat
109. Goat
110. Car
111. Goat
112. Goat
113. Goat
114. Goat
115. Goat
116. Goat
117. Car
118. Goat
119. Car
120. Car
121. Car
122. Goat
123. Goat
124. Goat
125. Car
126. Goat
127. Goat
128. Goat
129. Goat
130. Goat
131. Car
132. Goat
133. Goat
134. Goat
135. Car
136. Goat
137. Goat
138. Goat
139. Goat
140. Car
141. Car
142. Goat
143. Car
144. Goat
145. Goat
146. Car
147. Goat
148. Car
149. Goat
150. Goat
151. Car
152. Goat
153. Car
154. Car
155. Goat
156. Goat
157. Goat
158. Goat
159. Goat
160. Goat
161. Car
162. Goat
163. Car
164. Goat
165. Car
166. Car
167. Goat
168. Goat
169. Goat
170. Car
171. Goat
172. Goat
173. Goat
174. Goat
175. Goat
176. Car
177. Goat
178. Car
179. Car
180. Goat
181. Car
182. Goat
183. Goat
184. Goat
185. Goat
186. Goat
187. Car
188. Car
189. Car
190. Goat
191. Goat
192. Car
193. Goat
194. Car
195. Car
196. Car
197. Car
198. Goat
199. Goat
200. Goat
201. Goat
202. Car
203. Car
204. Goat
205. Car
206. Goat
207. Goat
208. Goat
209. Goat
210. Goat
211. Goat
212. Car
213. Goat
214. Goat
215. Goat
216. Goat
217. Goat
218. Car
219. Goat
220. Car
221. Goat
222. Goat
223. Goat
224. Goat
225. Goat
226. Goat
227. Goat
228. Goat
229. Goat
230. Goat
231. Goat
232. Goat
233. Goat
234. Goat
235. Goat
236. Goat
237. Goat
238. Goat
239. Goat
240. Goat
241. Goat
242. Goat
243. Goat
244. Goat
245. Goat
246. Car
247. Goat
248. Car
249. Goat
250. Car
251. Car
252. Goat
253. Car
254. Goat
255. Goat
256. Car
257. Car
258. Car
259. Goat
260. Goat
261. Car
262. Car
263. Car
264. Goat
265. Goat
266. Goat
267. Goat
268. Goat
269. Goat
270. Goat
271. Goat
272. Goat
273. Car
274. Goat
275. Car
276. Goat
277. Car
278. Goat
279. Car
280. Car
281. Goat
282. Goat
283. Goat
284. Goat
285. Goat
286. Car
287. Car
288. Car
289. Car
290. Goat
291. Goat
292. Car
293. Goat
294. Goat
295. Goat
296. Car
297. Goat
298. Goat
299. Goat
300. Goat
301. Goat
302. Car
303. Goat
304. Car
305. Goat
306. Car
307. Goat
308. Goat
309. Goat
310. Car
311. Goat
312. Goat
313. Goat
314. Car
315. Goat
316. Car
317. Car
318. Goat
319. Goat
320. Car
321. Goat
322. Goat
323. Car
324. Car
325. Goat
326. Goat
327. Car
328. Car
329. Car
330. Goat
331. Goat
332. Goat
333. Goat
334. Goat
335. Car
336. Goat
337. Goat
338. Car
339. Car
340. Goat
341. Goat
342. Car
343. Car
344. Car
345. Car
346. Car
347. Goat
348. Car
349. Car
350. Goat
351. Goat
352. Car
353. Car
354. Goat
355. Goat
356. Goat
357. Goat
358. Goat
359. Goat
360. Goat
361. Goat
362. Goat
363. Goat
364. Goat
365. Goat
366. Goat
367. Car
368. Goat
369. Car
370. Car
371. Goat
372. Goat
373. Goat
374. Car
375. Goat
376. Goat
377. Goat
378. Car
379. Car
380. Goat
381. Goat
382. Goat
383. Goat
384. Car
385. Goat
386. Goat
387. Goat
388. Goat
389. Goat
390. Goat
391. Goat
392. Goat
393. Goat
394. Goat
395. Car
396. Car
397. Goat
398. Goat
399. Car
400. Goat
401. Goat
402. Car
403. Car
404. Car
405. Goat
406. Goat
407. Goat
408. Car
409. Goat
410. Car
411. Goat
412. Car
413. Goat
414. Car
415. Car
416. Car
417. Goat
418. Goat
419. Car
420. Goat
421. Goat
422. Goat
423. Car
424. Car
425. Goat
426. Car
427. Goat
428. Car
429. Goat
430. Goat
431. Goat
432. Car
433. Car
434. Goat
435. Goat
436. Car
437. Goat
438. Goat
439. Goat
440. Goat
441. Car
442. Goat
443. Goat
444. Goat
445. Car
446. Goat
447. Car
448. Goat
449. Car
450. Goat
451. Goat
452. Goat
453. Goat
454. Goat
455. Goat
456. Car
457. Car
458. Car
459. Goat
460. Goat
461. Goat
462. Goat
463. Car
464. Goat
465. Goat
466. Car
467. Goat
468. Goat
469. Car
470. Goat
471. Goat
472. Goat
473. Goat
474. Car
475. Goat
476. Goat
477. Goat
478. Car
479. Goat
480. Goat
481. Car
482. Car
483. Goat
484. Goat
485. Goat
486. Goat
487. Goat
488. Car
489. Car
490. Goat
491. Car
492. Goat
493. Goat
494. Goat
495. Goat
496. Goat
497. Goat
498. Goat
499. Goat
500. Goat
501. Car
502. Goat
503. Goat
504. Car
505. Goat
506. Goat
507. Goat
508. Goat
509. Goat
510. Car
511. Goat
512. Goat
513. Car
514. Goat
515. Car
516. Goat
517. Goat
518. Goat
519. Goat
520. Goat
521. Car
522. Car
523. Goat
524. Goat
525. Goat
526. Goat
527. Goat
528. Goat
529. Goat
530. Goat
531. Goat
532. Goat
533. Goat
534. Goat
535. Goat
536. Car
537. Goat
538. Goat
539. Goat
540. Goat
541. Goat
542. Goat
543. Car
544. Goat
545. Car
546. Car
547. Goat
548. Goat
549. Car
550. Goat
551. Goat
552. Goat
553. Car
554. Goat
555. Goat
556. Car
557. Car
558. Goat
559. Goat
560. Goat
561. Goat
562. Car
563. Goat
564. Car
565. Goat
566. Goat
567. Car
568. Car
569. Car
570. Goat
571. Car
572. Goat
573. Goat
574. Goat
575. Goat
576. Goat
577. Goat
578. Car
579. Car
580. Car
581. Car
582. Goat
583. Car
584. Goat
585. Goat
586. Goat
587. Goat
588. Goat
589. Goat
590. Goat
591. Goat
592. Goat
593. Car
594. Goat
595. Goat
596. Goat
597. Car
598. Goat
599. Goat
600. Goat
601. Car
602. Goat
603. Goat
604. Goat
605. Goat
606. Car
607. Car
608. Goat
609. Goat
610. Goat
611. Goat
612. Car
613. Goat
614. Goat
615. Car
616. Car
617. Car
618. Goat
619. Goat
620. Goat
621. Goat
622. Goat
623. Goat
624. Car
625. Car
626. Car
627. Car
628. Car
629. Car
630. Goat
631. Goat
632. Goat
633. Goat
634. Car
635. Goat
636. Goat
637. Goat
638. Goat
639. Goat
640. Car
641. Goat
642. Car
643. Goat
644. Goat
645. Car
646. Goat
647. Car
648. Goat
649. Goat
650. Goat
651. Goat
652. Goat
653. Goat
654. Goat
655. Car
656. Goat
657. Goat
658. Goat
659. Goat
660. Goat
661. Car
662. Car
663. Goat
664. Goat
665. Car
666. Goat
667. Car
668. Goat
669. Goat
670. Car
671. Car
672. Goat
673. Car
674. Goat
675. Car
676. Goat
677. Car
678. Goat
679. Goat
680. Car
681. Goat
682. Goat
683. Goat
684. Goat
685. Goat
686. Goat
687. Goat
688. Goat
689. Car
690. Car
691. Car
692. Goat
693. Car
694. Car
695. Goat
696. Goat
697. Goat
698. Goat
699. Car
700. Goat
701. Goat
702. Goat
703. Goat
704. Goat
705. Goat
706. Car
707. Goat
708. Goat
709. Goat
710. Goat
711. Goat
712. Goat
713. Car
714. Goat
715. Goat
716. Car
717. Car
718. Goat
719. Goat
720. Goat
721. Car
722. Goat
723. Car
724. Goat
725. Car
726. Goat
727. Goat
728. Goat
729. Goat
730. Goat
731. Goat
732. Goat
733. Car
734. Car
735. Goat
736. Goat
737. Goat
738. Goat
739. Goat
740. Goat
741. Goat
742. Car
743. Goat
744. Goat
745. Goat
746. Goat
747. Goat
748. Car
749. Goat
750. Goat
751. Goat
752. Goat
753. Goat
754. Goat
755. Goat
756. Goat
757. Car
758. Car
759. Goat
760. Car
761. Goat
762. Goat
763. Goat
764. Goat
765. Car
766. Car
767. Goat
768. Goat
769. Car
770. Goat
771. Goat
772. Car
773. Goat
774. Car
775. Car
776. Car
777. Goat
778. Goat
779. Goat
780. Goat
781. Car
782. Goat
783. Car
784. Goat
785. Goat
786. Car
787. Goat
788. Car
789. Goat
790. Car
791. Goat
792. Car
793. Car
794. Car
795. Car
796. Goat
797. Goat
798. Goat
799. Goat
800. Goat
801. Car
802. Goat
803. Goat
804. Goat
805. Car
806. Goat
807. Goat
808. Car
809. Goat
810. Goat
811. Goat
812. Goat
813. Goat
814. Goat
815. Goat
816. Goat
817. Goat
818. Goat
819. Goat
820. Car
821. Goat
822. Car
823. Goat
824. Car
825. Car
826. Goat
827. Car
828. Goat
829. Goat
830. Goat
831. Goat
832. Goat
833. Car
834. Car
835. Car
836. Goat
837. Car
838. Goat
839. Goat
840. Goat
841. Goat
842. Goat
843. Goat
844. Goat
845. Car
846. Car
847. Car
848. Goat
849. Car
850. Car
851. Goat
852. Goat
853. Goat
854. Car
855. Goat
856. Goat
857. Goat
858. Goat
859. Car
860. Goat
861. Goat
862. Goat
863. Goat
864. Goat
865. Goat
866. Goat
867. Goat
868. Goat
869. Goat
870. Goat
871. Goat
872. Car
873. Car
874. Goat
875. Car
876. Goat
877. Car
878. Goat
879. Goat
880. Goat
881. Goat
882. Goat
883. Goat
884. Goat
885. Goat
886. Goat
887. Goat
888. Goat
889. Goat
890. Car
891. Goat
892. Goat
893. Goat
894. Goat
895. Goat
896. Car
897. Goat
898. Car
899. Goat
900. Goat
901. Goat
902. Goat
903. Car
904. Goat
905. Car
906. Car
907. Car
908. Car
909. Car
910. Car
911. Goat
912. Goat
913. Goat
914. Goat
915. Goat
916. Goat
917. Car
918. Goat
919. Car
920. Goat
921. Car
922. Goat
923. Goat
924. Goat
925. Car
926. Goat
927. Car
928. Goat
929. Goat
930. Goat
931. Car
932. Goat
933. Goat
934. Car
935. Car
936. Goat
937. Car
938. Goat
939. Goat
940. Goat
941. Goat
942. Goat
943. Goat
944. Car
945. Car
946. Goat
947. Goat
948. Goat
949. Goat
950. Car
951. Goat
952. Goat
953. Goat
954. Goat
955. Goat
956. Goat
957. Goat
958. Goat
959. Goat
960. Goat
961. Goat
962. Goat
963. Goat
964. Goat
965. Goat
966. Car
967. Goat
968. Goat
969. Goat
970. Car
971. Goat
972. Car
973. Car
974. Goat
975. Car
976. Goat
977. Goat
978. Goat
979. Goat
980. Goat
981. Goat
982. Goat
983. Car
984. Goat
985. Goat
986. Goat
987. Car
988. Goat
989. Car
990. Goat
991. Goat
992. Goat
993. Goat
994. Goat
995. Goat
996. Goat
997. Goat
998. Goat
999. Goat
1000. Car

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 109 times and lost 891 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 31 times and lost 69 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}