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. Car
6. Car
7. Goat
8. Goat
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 329 times and lost 671 times.

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