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. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Car
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 339 times and lost 661 times.

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

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 98 times and lost 902 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 32 times and lost 68 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;
	}
}