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. Car
4. Goat
5. Goat
6. Goat
7. Car
8. Car
9. Goat
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 343 times and lost 657 times.

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